Abstract
The ultimate goal of scientific research is the development of theories or models to explain and/or to predict different kinds of phenomena (Woodward 2003). As discussed in the previous chapters, research into students’ learning of mathematics helps to predict what they may learn about a specific mathematical concept and the conditions by which that learning takes place. This is an important part of mathematics education as a research field and it is one of the roles of APOS Theory.
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Notes
- 1.
Some researchers use the term “cognitive path” to describe a specific ordering of concepts that students seem to follow when learning a mathematical topic. Cognitive paths are found by means of a specific statistical method using data from students (Vinner and Hershkowitz 1980). A cognitive path describes a process of learning focused on the mathematical aspects of the concept. Although it may seem that there is some similarity between “cognitive path” and “genetic decomposition,” their focus and content are different.
Cognitive paths describe or suggest a linear cognitive progression based on an analysis of the mathematical aspects of the concepts involved. Instead of a linear progression, a genetic decomposition in APOS Theory describes the mental structures and the mechanisms by which those structures are constructed. Confusion with cognitive paths may explain some of the errors discussed later in this chapter.
- 2.
A diffraction grating is an instrument used to analyze the light coming from stars. It decomposes the incoming light by diffraction to obtain a pattern of colored lines. These patterns allow researchers to know the chemical composition of the star.
- 3.
All the names of interview subjects are pseudonyms. The interviewer’s words are identified with “I:” throughout the text. “I” does not denote a single individual, as there were different interviewers for different studies, and sometimes multiple interviewers for the same study or even the same interview.
- 4.
Here, \( K \) refers to a field, \( V \) stands for a set of tuples, \( \mathrm{ va} \) denotes addition defined on \( V \), and \( \mathrm{ sm} \) represents the scalar multiplication operation defined on \( K \) and \( V \).
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Arnon, I. et al. (2014). Genetic Decomposition. In: APOS Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7966-6_4
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